The goal of this project is to develop new radiofrequency (RF) gradient encoding methods to spatially encode signals in MRI. The methods would enable silent, low-cost MRI systems, leading to a substantial reduction in the cost of imaging and improved patient compliance and comfort. In conventional MRI, a received signal is localized to its spatial location of origin based on its temporal frequency, which is controlled using magnetic fields that are parallel to the main (B0) field of the scanner and vary linearly across space. There are many problems with these B0 gradient fields: they are loud and induce peripheral nerve stimulation, compromising patient comfort; they have relatively long switching times due to the high inductance of the coils; they require bulky cooling systems and customized amplifiers; and they are expensive, representing around 20-25% of the cost of a clinical scanner. A potential solution to these problems is to replace B0 gradients with RF gradients, which are silent and low-cost. Unfortunately, in spite of its potential RF gradient encoding has not yet become a clinical or commercial success. This is largely due to the fact that no existing RF gradient encoding method offers the orthogonality between contrast development and spatial encoding that is en- joyed by B0 gradients, or a straightforward path to convert existing B0 gradient-based MRI acquisition techniques to use RF encoding. The methods proposed in this project will be the first to meet these requirements, and will thus represent the first truly viable RF gradient-based imaging methods. The central innovation of this project is to use the Bloch-Siegert shift to spatially encode the MRI signal. As with B0 gradients, this encoding mechanism is based on the application of phase shifts to magnetization directly in the transverse plane, and therefore does not modulate the magnitude of the transverse magnetization, leaving image contrast unaffected by spatial encoding. The first Aim of the project is to develop new RF gradient coils and other RF hardware to enable 2D and 3D Cartesian imaging at 0.5 Tesla, including hardware strategies for simultaneous RF transmission and reception to enable frequency encoding by Bloch-Siegert shift. The second Aim is to develop new RF-encoded pulse sequences based on the Bloch-Siegert shift, leveraging recent innovations in RF pulse design for Bloch-Siegert phase encoding, and in RF pulse design for RF gradient-based slice-selective excitation. The third Aim is to develop robust algorithms to reconstruct images from RF-encoded data. Successful completion of these Aims would broadly prove the feasibility of the proposed RF encoding methods for MR imaging, paving the way for translation to humans.